Matrix equivalence

Last updated September 01, 2024

In linear algebra, two rectangular m-by-n matrices A and B are called equivalent if

The notion of equivalence should not be confused with that of similarity, which is only defined for square matrices, and is much more restrictive (similar matrices are certainly equivalent, but equivalent square matrices need not be similar). That notion corresponds to matrices representing the same endomorphism V → V under two different choices of a single basis of V, used both for initial vectors and their images.

Properties

Matrix equivalence is an equivalence relation on the space of rectangular matrices.

For two rectangular matrices of the same size, their equivalence can also be characterized by the following conditions

The matrices can be transformed into one another by a combination of elementary row and column operations.
Two matrices are equivalent if and only if they have the same rank.

If matrices are row equivalent then they are also matrix equivalent. However, the converse does not hold; matrices that are matrix equivalent are not necessarily row equivalent. This makes matrix equivalence a generalization of row equivalence.^[1]

Canonical form

The rank property yields an intuitive canonical form for matrices of the equivalence class of rank $k$ as

${\begin{pmatrix}1&0&0&&\cdots &&0\\0&1&0&&\cdots &&0\\0&0&\ddots &&&&0\\\vdots &&&1&&&\vdots \\&&&&0&&\\&&&&&\ddots &\\0&&&\cdots &&&0\end{pmatrix}}$ ,

where the number of $1$ s on the diagonal is equal to $k$ . This is a special case of the Smith normal form, which generalizes this concept on vector spaces to free modules over principal ideal domains. Thus:

Theorem: Any mxn matrix of rank k is matrix equivalent to the mxn matrix that is all zeroes except that the first k diagonal entries are ones. ^[1]Corollary: Matrix equivalent classes are characterized by rank: two same-sided matrixes are matrix equivalent if and only if they have the same rank. ^[1]

2x2 matrices

2x2 matrices only have three possible ranks: zero, one, or two. This means all 2x2 matrices fit into one of three matrix equivalent classes:^[1]

${\begin{pmatrix}0&0\\0&0\\\end{pmatrix}}$ , ${\begin{pmatrix}1&0\\0&0\\\end{pmatrix}}$ , ${\begin{pmatrix}1&0\\0&1\\\end{pmatrix}}$

This means all 2x2 matrices are equivalent to one of these matrices. There is only one zero rank matrix, but the other two classes have infinitely many members; The representative matrices above are the simplest matrix for each class.

Matrix similarity

Matrix similarity is a special case of matrix equivalence. If two matrices are similar then they are also equivalent. However, the converse is not true.^[2] For example these two matrices are equivalent but not similar:

${\begin{pmatrix}1&0\\0&1\\\end{pmatrix}}$ , ${\begin{pmatrix}1&2\\0&3\\\end{pmatrix}}$

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References

1 2 3 4 Hefferon, Jim. Linear Algebra (4th ed.). pp. 270–272. This article incorporates textfrom this source, which is available under the CC BY-SA 3.0 license.
↑ Hefferon, Jim. Linear Algebra (4th ed.). p. 405. This article incorporates textfrom this source, which is available under the CC BY-SA 3.0 license.

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[:0-1] 1 2 3 4 Hefferon, Jim. Linear Algebra (4th ed.). pp. 270–272. This article incorporates textfrom this source, which is available under the CC BY-SA 3.0 license.

[2] Hefferon, Jim. Linear Algebra (4th ed.). p. 405. This article incorporates textfrom this source, which is available under the CC BY-SA 3.0 license.

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v t e Matrix classes
Explicitly constrained entries	Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal Bisymmetric Block-diagonal Block Block tridiagonal Boolean Cauchy Centrosymmetric Conference Complex Hadamard Copositive Diagonally dominant Diagonal Discrete Fourier Transform Elementary Equivalent Frobenius Generalized permutation Hadamard Hankel Hermitian Hessenberg Hollow Integer Logical Matrix unit Metzler Moore Nonnegative Pentadiagonal Permutation Persymmetric Polynomial Quaternionic Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Vandermonde Walsh Z
Constant	Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero
Conditions on eigenvalues or eigenvectors	Companion Convergent Defective Definite Diagonalizable Hurwitz Positive-definite Stieltjes
Satisfying conditions on products or inverses	Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Unimodular Unipotent Unitary Totally unimodular Weighing
With specific applications	Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Confusion Coxeter Distance Duplication and elimination Euclidean distance Fundamental (linear differential equation) Generator Gram Hessian Householder Jacobian Moment Payoff Pick Random Rotation Seifert Shear Similarity Symplectic Totally positive Transformation
Used in statistics	Centering Correlation Covariance Design Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used in graph theory	Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel adjacency Tutte
Used in science and engineering	Cabibbo–Kobayashi–Maskawa Density Fundamental (computer vision) Fuzzy associative Gamma Gell-Mann Hamiltonian Irregular Overlap S State transition Substitution Z (chemistry)
Related terms	Jordan normal form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Row echelon form Wronskian
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